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Your data matches 55 different statistics following compositions of up to 3 maps.
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Matching statistic: St000375
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 0
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 0
[+,-,+] => [1,3,2] => 0
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 0
[-,+,-] => [2,1,3] => 0
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,2,3] => 0
[-,3,2] => [2,1,3] => 0
[2,1,+] => [1,3,2] => 0
[2,1,-] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,+,1] => [2,1,3] => 0
[3,-,1] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 0
[+,-,+,+] => [1,3,4,2] => 0
[+,+,-,+] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 0
[-,-,+,-] => [3,1,2,4] => 0
[-,+,-,-] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => 0
[+,3,2,+] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => 0
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001513
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 0
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 0
[+,-,+] => [1,3,2] => 0
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 0
[-,+,-] => [2,1,3] => 0
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,2,3] => 0
[-,3,2] => [2,1,3] => 0
[2,1,+] => [1,3,2] => 0
[2,1,-] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,+,1] => [2,1,3] => 0
[3,-,1] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 0
[+,-,+,+] => [1,3,4,2] => 0
[+,+,-,+] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 0
[-,+,-,+] => [2,4,1,3] => 0
[-,+,+,-] => [2,3,1,4] => 0
[+,-,-,+] => [1,4,2,3] => 0
[+,-,+,-] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 0
[-,-,+,-] => [3,1,2,4] => 0
[-,+,-,-] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => 0
[+,-,4,3] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => 0
[+,3,2,+] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => 0
[+,3,2,-] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => 0
Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St000233
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => {{1}}
=> 0
[-] => [1] => {{1}}
=> 0
[+,+] => [1,2] => {{1},{2}}
=> 0
[-,+] => [2,1] => {{1,2}}
=> 0
[+,-] => [1,2] => {{1},{2}}
=> 0
[-,-] => [1,2] => {{1},{2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 0
[+,+,+] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,+,+] => [2,3,1] => {{1,2,3}}
=> 0
[+,-,+] => [1,3,2] => {{1},{2,3}}
=> 0
[+,+,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,-,+] => [3,1,2] => {{1,3},{2}}
=> 0
[-,+,-] => [2,1,3] => {{1,2},{3}}
=> 0
[+,-,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,-,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[+,3,2] => [1,2,3] => {{1},{2},{3}}
=> 0
[-,3,2] => [2,1,3] => {{1,2},{3}}
=> 0
[2,1,+] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,-] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,1,2] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,+,1] => [2,1,3] => {{1,2},{3}}
=> 0
[3,-,1] => [1,3,2] => {{1},{2,3}}
=> 0
[+,+,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,+,+,+] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[+,-,+,+] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[+,+,-,+] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[+,+,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,+,+] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
[-,+,-,+] => [2,4,1,3] => {{1,2,4},{3}}
=> 0
[-,+,+,-] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[+,-,-,+] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[+,-,+,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[+,+,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,-,+] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
[-,-,+,-] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
[-,+,-,-] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[+,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[+,+,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,+,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[+,-,4,3] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[-,-,4,3] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
[+,3,2,+] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[-,3,2,+] => [2,4,1,3] => {{1,2,4},{3}}
=> 0
[+,3,2,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,3,2,-] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[+,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[-,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[+,4,2,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St001550
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [2,1] => 0
[-,+] => [2,1] => [1,2] => 0
[+,-] => [1,2] => [2,1] => 0
[-,-] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[+,+,+] => [1,2,3] => [2,3,1] => 0
[-,+,+] => [2,3,1] => [3,1,2] => 0
[+,-,+] => [1,3,2] => [2,1,3] => 0
[+,+,-] => [1,2,3] => [2,3,1] => 0
[-,-,+] => [3,1,2] => [1,3,2] => 0
[-,+,-] => [2,1,3] => [3,2,1] => 0
[+,-,-] => [1,2,3] => [2,3,1] => 0
[-,-,-] => [1,2,3] => [2,3,1] => 0
[+,3,2] => [1,2,3] => [2,3,1] => 0
[-,3,2] => [2,1,3] => [3,2,1] => 0
[2,1,+] => [1,3,2] => [2,1,3] => 0
[2,1,-] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => 0
[3,+,1] => [2,1,3] => [3,2,1] => 0
[3,-,1] => [1,3,2] => [2,1,3] => 0
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => 0
[-,+,+,+] => [2,3,4,1] => [3,4,1,2] => 0
[+,-,+,+] => [1,3,4,2] => [2,4,1,3] => 0
[+,+,-,+] => [1,2,4,3] => [2,3,1,4] => 0
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,+,+] => [3,4,1,2] => [4,1,3,2] => 0
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 0
[-,+,+,-] => [2,3,1,4] => [3,4,2,1] => 0
[+,-,-,+] => [1,4,2,3] => [2,1,4,3] => 0
[+,-,+,-] => [1,3,2,4] => [2,4,3,1] => 0
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,-,+] => [4,1,2,3] => [1,3,4,2] => 0
[-,-,+,-] => [3,1,2,4] => [4,2,3,1] => 0
[-,+,-,-] => [2,1,3,4] => [3,2,4,1] => 0
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => 0
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => 0
[+,+,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[-,+,4,3] => [2,3,1,4] => [3,4,2,1] => 0
[+,-,4,3] => [1,3,2,4] => [2,4,3,1] => 0
[-,-,4,3] => [3,1,2,4] => [4,2,3,1] => 0
[+,3,2,+] => [1,2,4,3] => [2,3,1,4] => 0
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => 0
[+,3,2,-] => [1,2,3,4] => [2,3,4,1] => 0
[-,3,2,-] => [2,1,3,4] => [3,2,4,1] => 0
[+,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0
[-,3,4,2] => [2,1,3,4] => [3,2,4,1] => 0
[+,4,2,3] => [1,2,3,4] => [2,3,4,1] => 0
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001549
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[+,-,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,1,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,-,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,-,4,3] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,3,2,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St000205
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 95%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 95%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[-,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[+,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[-,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[2,1] => [2,1] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[-,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[+,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[-,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[+,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[-,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[+,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[-,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,+,+] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,-,+] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,+,-] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,-,-] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,+] => [2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,1,-] => [2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [2,3,4,1] => [3,1]
=> [1]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,2,+] => [3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,2,-] => [3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,1]
=> [1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,+,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,-,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,-,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,+,-,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,-,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 95%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 95%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[-,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[+,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[-,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,0,0}
[2,1] => [2,1] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[-,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 0
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[-,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[+,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[-,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[+,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[-,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[+,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[-,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,+,+] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,-,+] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,+,-] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,-,-] => [2,1,3,4] => [3,1]
=> [1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,+] => [2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,1,-] => [2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [2,3,4,1] => [3,1]
=> [1]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,2,+] => [3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,2,-] => [3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,1]
=> [1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,+,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,+,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,-,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,-,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,-,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[+,+,+,-,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,-,+,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,+,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,-,+,+,-] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[-,+,-,-,+] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St001175
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,0}
[-] => [1] => [1]
=> []
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,4,5,1] => [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,5,1,4] => [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,5,3] => [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,5,3,1] => [2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,1,3,4] => [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,5,2] => [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,5,2,4] => [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,5,1] => [3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,5,1,2] => [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,5,3] => [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,5,3,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,1,5,2] => [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,5,2,1] => [4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,1,2,3] => [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,2,3,1] => [4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,2,3,4] => [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,4,2,3] => [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,1,2,4] => [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,4,1,2] => [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,1,3,2] => [5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,2,1,3] => [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,6,1,5] => [2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,1,6,4] => [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,5,6,4,1] => [2,3,5,6,4,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,6,1,4,5] => [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,6,5,1,4] => [2,3,6,5,1,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,5,6,3] => [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,6,3,5] => [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,5,3,6,1] => [2,4,5,3,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,5,6,1,3] => [2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,6,3,1,5] => [2,4,6,3,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,6,5,3,1] => [2,4,6,5,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,1,3,6,4] => [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,1,6,4,3] => [2,5,1,6,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,4,1,6,3] => [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001498
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1,0]
=> [1,1,0,0]
=> ? ∊ {0,0}
[-] => [1] => [1,0]
=> [1,1,0,0]
=> ? ∊ {0,0}
[+,+] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[-,+] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[+,-] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[-,-] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> ? = 0
[+,+,+] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,+,+] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,-,+] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,+,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,-,+] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,+,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,-,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,-,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,+] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,-] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0}
[3,+,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0}
[3,-,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0}
[+,+,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[+,+,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[-,+,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[+,-,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[-,-,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[+,3,2,+] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[-,3,2,+] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[+,3,2,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[-,3,2,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[+,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[-,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[+,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[-,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[+,4,+,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[-,4,+,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[+,4,-,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[-,4,-,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,1,+,+] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,1,+,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,1,-,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,+,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,-,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,+,+,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,-,+,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,+,-,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,-,-,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[5,1,2,3,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,2,+,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,2,-,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,+,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,-,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,+,+,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,-,+,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,+,-,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,-,-,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,4,2,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,1,+,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,1,+,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,1,-,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,1,-,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,+,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,+,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,-,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,-,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,+,+,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,+,+,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,-,+,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,+,-,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,-,+,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,+,-,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,-,-,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,-,-,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,+,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,-,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001570
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => ([],1)
=> ([],1)
=> ? ∊ {0,0}
[-] => [1] => ([],1)
=> ([],1)
=> ? ∊ {0,0}
[+,+] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[-,+] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[+,-] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[-,-] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[+,+,+] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,1,+] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,1,-] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[3,+,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,-,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,+,+,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,+,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,+,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,+,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,-,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-,+] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,+,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,-,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,-,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,-,-] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[-,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[+,4,+,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,+,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,-,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,-,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,+,+] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,4,+,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,-,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,+,1,+] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,-,1,+] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,+,1,-] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,-,1,-] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,+,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,-,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,+,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,-,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,+,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,-,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,+,+,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,-,+,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,+,-,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,-,-,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,+,2,-] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,-,2,+] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,+,2,-] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,-,2,-] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,-,2,-] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,+,5,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,+,5,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,-,5,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[-,4,-,5,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[+,4,5,3,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000455The second largest eigenvalue of a graph if it is integral. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001890The maximum magnitude of the Möbius function of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St001429The number of negative entries in a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001964The interval resolution global dimension of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000181The number of connected components of the Hasse diagram for the poset. St000907The number of maximal antichains of minimal length in a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
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